Introduction

 
Creation of a Cohomology Module
      CohomologyModule(G, M) : GrpPerm, ModGrp -> ModCoho
      CohomologyModule(G, Q, T) : GrpPerm, SeqEnum, SeqEnum -> ModCoho
      Example GrpCoh_coho-module1 (H75E1)
      Example GrpCoh_coho-module4 (H75E2)
      Example GrpCoh_coho-module2 (H75E3)
      Example GrpCoh_coho-module3 (H75E4)
      CohomologyModule(G, A, M) : GrpPerm, GrpAb, Any -> ModCoho

 
Accessing Properties of the Cohomology Module
      Module(CM) : ModCoho -> ModGrp
      Invariants(CM) : ModCoho -> SeqEnum
      Dimension(CM) : ModCoho -> RngIntElt
      Ring(CM) : ModCoho -> ModGrp
      Group(CM) : ModCoho -> Grp
      FPGroup(CM) : ModCoho -> Grp, HomGrp
      MatrixOfElement(CM, g) : ModCoho, GrpElt -> AlgMatElt
      Example GrpCoh_coho-module2cont (H75E5)

 
Calculating Cohomology
      CohomologyGroup(CM, n) : ModCoho, RngIntElt -> ModTupRng
      Example GrpCoh_coho-module2cont (H75E6)
      Example GrpCoh_coho-module3cont (H75E7)
      CohomologicalDimension(CM, n) : ModCoho, RngIntElt -> RngIntElt
      CohomologicalDimension(M, n) : ModGrp, n -> RngIntElt
      CohomologicalDimensions(M, n) : ModGrp, n -> RngIntElt
      CohomologicalDimension(G, M, n) : GrpPerm, ModRng, RngIntElt -> RngIntElt
      H1Dimension(F, f, M) : GrpFP, Map, ModGrp -> RngIntElt
      H1Dimension(G, f, K) : GrpFP, Map, Rng -> RngIntElt
      H1DimensionSymmetricSquare(G, f, K) : GrpFP, Map, Rng -> RngIntElt
      Example GrpCoh_coho-example (H75E8)
      Example GrpCoh_coho-module4-cont (H75E9)
      Example GrpCoh_more-difficult (H75E10)

 
Cocycles
      ZeroCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
      IdentifyZeroCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
      OneCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
      IdentifyOneCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
      IsOneCoboundary(CM, s) : ModCoho, UserProgram -> BoolElt, UserProgram
      TwoCocycle(CM, s) : ModCoho, SeqEnum -> UserProgram
      IdentifyTwoCocycle(CM, s) : ModCoho, UserProgram -> ModTupRngElt
      IsTwoCoboundary(CM, s) : ModCoho, UserProgram -> BoolElt, UserProgram
      Example GrpCoh_cocycles (H75E11)

 
The Restriction to a Subgroup
      Restriction(CM, H) : ModCoho, Grp -> ModCoho
      Example GrpCoh_restriction (H75E12)

 
Other Operations on Cohomology Modules
      CorestrictionMapImage(G, C, c, i) : Grp, ModCoho, UserProgram, RngIntElt -> UserProgram
      InflationMapImage(M, c) : Map, UserProgram -> UserProgram
      CoboundaryMapImage(M, i, c) : ModCoho, RngIntElt, UserProgram -> UserProgram

 
Constructing Extensions
      Extension(CM, s) : ModCoho, SeqEnum -> Grp, HomGrp, Map
      Extension(GrpPerm, CM, s) : Cat, ModCoho, SeqEnum -> GrpPerm, HomGrp, Map
      Example GrpCoh_A7cover (H75E13)
      Example GrpCoh_Dempwolff (H75E14)
      SplitExtension(CM) : ModCoho -> Grp, HomGrp, Map
      SplitExtension(G, M) : ModCoho -> Grp, ModGrp -> HomGrp, Map
      SplitExtension(GrpPerm,CM) : Cat, ModCoho -> GrpPerm, HomGrp, Map
      Example GrpCoh_split-extension (H75E15)
      Example GrpCoh_split-extension (H75E16)
      pMultiplicator(G, p) : GrpPerm, RngIntElt -> [ RngIntElt ]
      pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFP
      Example GrpCoh_straightforward (H75E17)
      Example GrpCoh_nonsplit $2^5.L_5(2) (H75E18)
      Example GrpCoh_module-integers (H75E19)

 
Constructing Distinct Extensions
      DistinctExtensions(CM) : ModCoho -> SeqEnum
      Example GrpCoh_distinct-extensions (H75E20)
      ExtensionsOfElementaryAbelianGroup(p, d, G) : RngIntElt, RngIntElt, GrpPerm -> SeqEnum
      Example GrpCoh_extensions-abelian (H75E21)
      ExtensionsOfSolubleGroup(H, G) : GrpPerm, GrpPerm -> SeqEnum
      Example GrpCoh_extensions-soluble (H75E22)
      Example GrpCoh_distinct-extensions (H75E23)
      IsExtensionOf(G) : GrpPerm -> [],
      IsExtensionOf(L) : [GrpPerm] -> [], []

 
Finite Group Cohomology

      Creation of Gamma-groups
            GammaGroup(Gamma, A, action) : Grp, Grp, Map[Grp, GrpAuto] -> GGrp
            InducedGammaGroup(A, B) : GGrp, Grp -> GGrp
            Example GrpCoh_createGGrp (H75E24)
            IsNormalised(B, action) : Grp, Map -> BoolElt
            IsInduced(AmodB) : GGrp -> BoolElt, GGrp, GGrp, Map, Map

      Accessing Information
            Group(A) : GGrp -> Grp
            GammaAction(A) : GGrp -> Map[Grp, GrpAuto]
            ActingGroup(A) : GGrp -> Grp

      One Cocycles
            OneCocycle(A, imgs) : GGrp, SeqEnum[GrpElt] -> OneCoC
            TrivialOneCocycle(A) : GGrp -> OneCoC
            IsOneCocycle(A, imgs) : GGrp, SeqEnum[GrpElt] -> BoolElt, OneCoC
            AreCohomologous(alpha, beta) : OneCoC, OneCoC -> BoolElt, GrpElt
            CohomologyClass(alpha) : OneCoC -> SetIndx[OneCoC]
            InducedOneCocycle(AmodB, alpha) : GGrp, OneCoC -> OneCoC
            ExtendedOneCocycle(alpha) : OneCoC -> SetEnum[OneCoC]
            ExtendedCohomologyClass(alpha) : OneCoC -> SetEnum[OneCoC]
            GammaGroup(alpha) : OneCoC -> GGrp
            CocycleMap(alpha) : OneCoC -> Map

      Group Cohomology
            Cohomology(A, n) : GGrp, RngIntElt -> SetEnum[OneCoC]
            OneCohomology(A) : GGrp -> SetEnum[OneCoC]
            TwistedGroup(A, alpha) : GGrp, OneCoC -> GGrp
            Example GrpCoh_large example (H75E25)

 
Bibliography

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